An instanton is a field configuration that vanishes at spacetime infinity. It is the quantum effect that leads metastable states to decay into vacuum. It is a phenomenon that takes place in usual spacetime with signature \((-,+,+,+)\)  but in order to perform physical calculation we use its equivalence with a soliton solution (static and energetically stable field configuration) in Euclidean spacetime.
In Yang–Mills theory, self-duality of the field strength \(F_{\mu \nu }=\epsilon _{\mu \nu \rho \sigma }F^{\rho \sigma }\)  in four spacetime dimensions is a widely known instanton configuration [1]}. Similarly, self-duality constraint on the curvature two-form (and connection 1-form) in Cartan's formalism of general relativity characterizes a gravitational instanton. An important feature of self-duality of the curvature is that the Ricci-tensor vanishes and it is a solution of the vacuum Einstein equations.
Also, self-dual curvature leads to solving a linear differential equation, a task much easier than solving the full non-linear Einstein equations. Gravitational instantons were found in Bianchi IX metrics, by Gibbons and Pope [2]}. Without realizing it, they arrived at Darboux–Halphen system from self-duality constraints.
